Mean-field interactions between living cells in linear and nonlinear elastic matrices

Living cells respond to mechanical changes in the matrix surrounding them by applying contractile forces that are in turn transmitted to distant cells. We consider simple effective geometries for the spatial arrangement of cells, we calculate the mechanical work that each cell performs in order to deform the matrix, and study how that energy changes when a contracting cell is surrounded by other cells with similar properties and behavior. Cells regulating the displacements that they generate are attracted to each other in a manner that does not depend on the cell's rigidity. Whereas cells regulating the active stress that they apply repel each other. This repulsion depends on the cell's bulk modulus in spherical geometry, while in cylindrical geometries the interaction depends also on their shear modulus. In nonlinear, strain-stiffening matrices, for displacement regulation, in the presence of other cells, cell contraction is limited due to the divergence of the shear stress. For stress regulation, the interaction energy drops at the nonlinear stiffening stress. Our theoretical work provides insight into matrix-mediated interactions between contractile cells and on the role of their mechanical regulatory behavior.

Figure 1

Schematic of a single cell in an infinite matrix (a) and in a bounded region (b). The cell may be a sphere (c), a flat cylinder (d) for plane stress, or a long cylinder (e) for plane strain.

Figure 2

Stress stiffenning for simple shear: (a) Stress vs strain is linear for small strain and then diverges at ${\gamma }_a=1/\sqrt{b}$. (b) Differential shear modulus is constant for small stress and then increases as $G\propto {\sigma }^{3/2}$ for stress larger than $\mu {\gamma }_a$.

Figure 3

Normalized displacement $\tilde{u}=u/u_0$ vs normalized radius $\tilde{r}=r/R_0$ in spherical nonlinear medium for the (a) unbounded geometry and (b) geometry bounded at ${\tilde{R}}_1=R_1/R_0=10,100,250,500,1000$. Colors mark the dimensionless strength $A=b{u}_0^2/R_0^2$ of the nonlinearity, as indicated in the legend. All results are for Poisson ratio $\nu =0.4$. In both geometries, for increasing nonlinearity, the near-field behavior tends toward the shearless solution $\tilde{u}=\tilde{r}$ represented by the blue dotted line and may be approximated by Eq. (53), which is drawn with a black dashed line only for the maximal $A$ value for each case.

Figure 4

(a) Maximal possible cell contraction increases with cell-cell distance according to Eq. (56). Numerical results for $b=0.1,0.5,0.8,1.1,2,5,10$ all collapse to a single curve, which perfectly agrees with the theoretical prediction. (b) Position of maximal displacement increases with dimensionless nonlinearity according to Eq. (59) both for the bounded and for the unbounded geometries. (c) Effective far-field displacement increases with $A$ and decreases with ${\tilde{R}}_1$ following Eq. (61). In all panels, theoretical prediction is marked with a red dotted line.

Figure 5

Interaction energy vs cell contraction for different values of $R_1$ and $b$. Interaction energy is scaled by $\mu R_0$, and cell displacement is scaled by $R_0$. Black dashed lines are linear elasticity behavior for $b=0$.

Figure 6

Scaled interaction energy behaves according to linear elasticity (15) for small cell contraction (black dashed line). As $u_0$ approaches $u_a$, the interaction energy grows according to Eq. (63) (red dashed lines), until it diverges at $u_0=u_a$ (blue dashed line). Results for different values of the material nonlinearity $b$ collapse to curves that depend only on ${\tilde{R}}_1$.

Figure 7

Crossover of $\tilde{u}\left(\tilde{r}\right)$ from near-field behavior of Eq. (53) (dashed black line) to far-field $\tilde{u}={\tilde{u}}_{\mathrm{eff}}/{\tilde{r}}^2$ behavior. Matching the two solutions at ${\tilde{R}}_{*}$ (green dashed line) underestimates ${\tilde{u}}_{\mathrm{eff}}$. Matching at a slightly larger distance (orange dashed line) is obtained by taking $\alpha >1$ in Eq. (61). Results are shown for $A=35.7858$.

Figure 8

Interaction energy with stress regulation in a nonlinear medium. (a–c) Results for different values of ${\tilde{R}}_1$ and $b$. Dashed black lines are linear elasticity behavior (20). Interaction energy is scaled by $\mu /R_0^3$ and stress is scaled by $\mu$. (d) When normalizing the interaction energy by the linear prediction and normalizing the stress by the typical stress $\mu {\gamma }_a$ where strain stiffening becomes dominant, results for all $b$ and ${\tilde{R}}_1$ collapse to a single curve.